What's string Theory

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CERN-TH/97-218 hep-th/9709062 INTRODUCTION TO SUPERSTRING THEORY ∗ Elias Kiritsis Theory Division, CERN, CH-1211, Geneva 23, SWITZERLAND Abstract In these lecture notes, an introduction to superstringtheory is presented. Classi- cal strings, covariant and light-cone quantization, supersymmetric strings, anomaly cancelation, compactification, T-duality, supersymmetry breaking, and threshold corrections to low-energy couplings are discussed. A brief introduction to non- perturbative duality symmetries is also included. Lectures presented at the Catholic University of Leuven and at the University of Padova during the academic year 1996-97. To be published by Leuven University Press. CERN-TH/97-218 March 1997 ∗ e-mail: KIRITSISNXTH04.CERN.CH arXiv:hep-th/9709062v2 30 Mar 1998Contents 1 Introduction 5 2 Historical perspective 6 3 Classical string theory 9 3.1 The point particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Relativistic strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 Oscillator expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4 Quantization of the bosonic string 23 4.1 Covariant canonical quantization . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Light-cone quantization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.3 Spectrum of the bosonic string . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.4 Path integral quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.5 Topologically non-trivial world-sheets . . . . . . . . . . . . . . . . . . . . . 30 4.6 BRST primer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.7 BRST in string theory and the physical spectrum . . . . . . . . . . . . . . 33 5 Interactions and loop amplitudes 36 6 Conformal field theory 38 6.1 Conformal transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 6.2 Conformally invariant field theory . . . . . . . . . . . . . . . . . . . . . . . 41 6.3 Radial quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.4 Example: the free boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6.5 The central charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6.6 The free fermion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.7 Mode expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.8 The Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.9 Representations of the conformal algebra . . . . . . . . . . . . . . . . . . . 54 6.10 Affine algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.11 Free fermions and O(N) affine symmetry . . . . . . . . . . . . . . . . . . . 60 16.12 N=1 superconformal symmetry . . . . . . . . . . . . . . . . . . . . . . . . 66 6.13 N=2 superconformal symmetry . . . . . . . . . . . . . . . . . . . . . . . . 68 6.14 N=4 superconformal symmetry . . . . . . . . . . . . . . . . . . . . . . . . 70 6.15 The CFT of ghosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7 CFT on the torus 75 7.1 Compact scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 7.2 Enhanced symmetry and the string Higgs effect . . . . . . . . . . . . . . . 84 7.3 T-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.4 Free fermions on the torus . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.5 Bosonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.6 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.7 CFT on higher-genus Riemann surfaces . . . . . . . . . . . . . . . . . . . . 97 8 Scattering amplitudes and vertex operators of bosonic strings 98 9 Strings in background fields and low-energy effective actions 102 10 Superstrings and supersymmetry 104 10.1 Closed (type-II) superstrings . . . . . . . . . . . . . . . . . . . . . . . . . . 106 10.2 Massless R-R states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 10.3 Type-I superstrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 10.4 Heterotic superstrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 10.5 Superstring vertex operators . . . . . . . . . . . . . . . . . . . . . . . . . . 117 10.6 Supersymmetric effective actions. . . . . . . . . . . . . . . . . . . . . . . . 119 11 Anomalies 122 12 Compactification and supersymmetry breaking 130 12.1 Toroidal compactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 12.2 Compactification on non-trivial manifolds . . . . . . . . . . . . . . . . . . 135 12.3 World-sheet versus spacetime supersymmetry . . . . . . . . . . . . . . . . 140 12.4 Heterotic orbifold compactifications with N=2 supersymmetry . . . . . . . 145 12.5 Spontaneous supersymmetry breaking . . . . . . . . . . . . . . . . . . . . . 153 212.6 Heterotic N=1 theories and chirality in four dimensions . . . . . . . . . . . 155 12.7 Orbifold compactifications of the type-II string . . . . . . . . . . . . . . . . 157 13 Loop corrections to effective couplings in string theory 159 13.1 Calculation of gauge thresholds . . . . . . . . . . . . . . . . . . . . . . . . 161 13.2 On-shell infrared regularization . . . . . . . . . . . . . . . . . . . . . . . . 166 13.3 Gravitational thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 13.4 Anomalous U(1)’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 13.5 N=1,2 examples of threshold corrections . . . . . . . . . . . . . . . . . . . 172 13.6 N=2 universality of thresholds . . . . . . . . . . . . . . . . . . . . . . . . . 175 13.7 Unification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 14 Non-perturbative string dualities: a foreword 179 14.1 Antisymmetric tensors and p-branes . . . . . . . . . . . . . . . . . . . . . . 183 14.2 BPS states and bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 14.3 Heterotic/type-I duality in ten dimensions. . . . . . . . . . . . . . . . . . . 186 14.4 Type-IIA versus M-theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 14.5 M-theory and the E ×E heterotic string . . . . . . . . . . . . . . . . . . . 196 8 8 14.6 Self-duality of the type-IIB string . . . . . . . . . . . . . . . . . . . . . . . 196 14.7 D-branes are the type-II R-R charged states . . . . . . . . . . . . . . . . . 199 14.8 D-brane actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 14.9 Heterotic/type-II duality in six and four dimensions . . . . . . . . . . . . . 205 15 Outlook 211 Acknowledgments 212 Appendix A: Theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Appendix B: Toroidal lattice sums . . . . . . . . . . . . . . . . . . . . . . . . . 216 Appendix C: Toroidal Kaluza-Klein reduction . . . . . . . . . . . . . . . . . . . 219 Appendix D: N=1,2,4, D=4 supergravity coupled to matter . . . . . . . . . . . 221 Appendix E: BPS multiplets and helicity supertrace formulae . . . . . . . . . . 224 Appendix F: Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Appendix G: Helicity string partition functions . . . . . . . . . . . . . . . . . . 234 3Appendix H: Electric-Magnetic duality in D=4 . . . . . . . . . . . . . . . . . . 240 References 243 41 Introduction String theory has been the leading candidate over the past years for a theory that consis- tently unifies all fundamental forces of nature, including gravity. In a sense, the theory predicts gravity and gauge symmetry around flat space. Moreover, the theory is UV- finite. The elementary objects are one-dimensional strings whose vibration modes should correspond to the usual elementary particles. At distanceslargewith respect tothesize ofthe strings, thelow-energyexcitations can be described by an effective field theory. Thus, contact can be established with quantum field theory, which turned outto be successful in describing the dynamics of the real world at low energy. I will try to explain here the basic structure of string theory, its predictions and prob- lems. In chapter 2 the evolution of string theory is traced, from a theory initially built to describehadronstoa“theoryofeverything”. Inchapter3adescriptionofclassicalbosonic stringtheoryisgiven. Theoscillationmodesofthestringaredescribed,preparingthescene forquantization. Inchapter4,thequantizationofthebosonicstringisdescribed. Allthree different quantization procedures are presented to varying depth, since in each one some specific properties are more transparent than in others. I thus describe the old covariant quantization, the light-cone quantization and the modern path-integral quantization. In chapter 6 a concise introduction is given, to the central concepts of conformal field theory since it is the basic tool in discussing first quantized string theory. In chapter 8 the calculation of scattering amplitudes is described. In chapter 9 the low-energy effective action for the massless modes is described. In chapter 10 superstrings are introduced. They provide spacetime fermions and real- ize supersymmetry in spacetime and on the world-sheet. I go through quantization again, and describe the different supersymmetric string theories in ten dimensions. In chapter 11 gauge and gravitational anomalies are discussed. In particular it is shown that the super- string theories are anomaly-free. In chapter 12 compactifications of the ten-dimensional superstring theories are described. Supersymmetry breaking is also discussed in this con- text. In chapter 13, I describe how to calculate loop corrections to effective coupling constants. This is very important for comparing string theory predictions at low energy with the real world. In chapter 14 a brief introduction to non-perturbative string con- nections and non-perturbative effects is given. This is a fast-changing subject and I have just included some basics as well as tools, so that the reader orients him(her)self in the web of duality connections. Finally, in chapter 15 a brief outlook and future problems are presented. Ihaveaddedanumberofappendicestomakeseveraltechnicaldiscussionsself-contained. 5In Appendix A useful information on the elliptic ϑ-functions is included. In Appendix B, I rederive the various lattice sums that appear in toroidal compactifications. In Appendix C the Kaluza-Klein ansatz is described, used to obtain actions in lower dimensions after toroidalcompactification. InAppendixDsomefactsarepresented aboutfour-dimensional locallysupersymmetric theorieswith N=1,2,4supersymmetry. InAppendix E,BPS states are described along with their representation theory and helicity supertrace formulae that can be used to trace their appearance in a supersymmetric theory. In Appendix F facts about elliptic modular forms are presented, which are useful in many contexts, notably in the one-loop computation of thresholds and counting of BPS multiplicities. In Ap- pendix G, I present the computation of helicity-generating string partition functions and the associated calculation of BPS multiplicities. Finally, in Appendix H, I briefly review electric–magnetic duality in four dimensions. I have not tried to be complete in my referencing. The focus was to provide, in most cases, appropriate reviews for further reading. Only in the last chapter, which covers very recent topics, I do mostly refer to original papers because of the scarcity of relevant reviews. 2 Historical perspective In the sixties, physicists tried to make sense of a big bulk of experimental data relevant to the strong interaction. There were lots of particles (or “resonances”) and the situation could best be described as chaotic. There were some regularities observed, though: • Almost linear Regge behavior. It was noticed that the large number of resonances could be nicely put on (almost) straight lines by plotting their mass versus their spin J 2 m = , (2.1) ′ α ′ −2 with α ∼1 GeV , and this relation was checked up to J = 11/2. • s-t duality. If we consider a scattering amplitude of two→ two hadrons (1,2→ 3,4), then it can be described by the Mandelstam invariants 2 2 2 s =−(p +p ) , t =−(p +p ) , u=−(p +p ) , (2.2) 1 2 2 3 1 3 P 2 with s+t+u= m . We are using a metric with signature (−+++). Such an ampli- i i tude depends on the flavor quantum numbers of hadrons (for example SU(3)). Consider the flavor part, which is cyclically symmetric in flavor space. For the full amplitude to be symmetric, it must also be cyclically symmetric in the momenta p . This symmetry i amounts to the interchange t ↔ s. Thus, the amplitude should satisfy A(s,t) = A(t,s). Consider a t-channel contribution due to the exchange of a spin-J particle of mass M. 6Then, at high energy J (−s) A (s,t)∼ . (2.3) J 2 t−M Thus, this partial amplitude increases with s and its behavior becomes worse for large values of J. If one sews amplitudes of this form together to make a loop amplitude, then there are uncontrollable UV divergences for J 1. Any finite sum of amplitudes of the form (2.3) has this bad UV behavior. However, if one allows an infinite number of terms then it is conceivable that the UV behavior might be different. Moreover such a finite sum has no s-channel poles. A proposal for such a dual amplitude was made by Veneziano 1 Γ(−α(s))Γ(−α(t)) A(s,t)= , (2.4) Γ(−α(s)−α(t)) where Γ is the standard Γ-function and ′ α(s)=α(0)+αs. (2.5) By using the standard properties of the Γ-function it can be checked that the amplitude (2.4) has an infinite number of s,t-channel poles: ∞ X (α(s)+1)...(α(s)+n) 1 A(s,t) =− . (2.6) n α(t)−n n=0 In this expansion the s ↔ t interchange symmetry of (2.4) is not manifest. The poles 2 in (2.6) correspond to the exchange of an infinite number of particles of mass M = ′ (n−α(0)/α) and high spins. It can also be checked that the high-energy behavior of the Veneziano amplitude is softer than any local quantum field theory amplitude, and the infinite number of poles is crucial for this. ItwassubsequentlyrealizedbyNambuandGotothatsuchamplitudescameoutofthe- ories of relativistic strings. However such theories had several shortcomings in explaining the dynamics of strong interactions. • All of them seemed to predict a tachyon. • Several of them seemed to contain a massless spin-2 particle that was impossible to get rid of. • All of them seemed to require a spacetime dimension of 26 in order not to break Lorentz invariance at the quantum level. • They contained only bosons. At the same time, experimental data from SLAC showed that at even higher energies hadrons have a point-like structure; this opened the way for quantum chromodynamics as the correct theory that describes strong interactions. 7However some workcontinued inthecontext of“dualmodels”andinthemid-seventies several interesting breakthroughs were made. •ItwasunderstoodbyNeveu, SchwarzandRamondhowtoincludespacetimefermions in string theory. •ItwasalsounderstoodbyGliozzi, Scherk andOlivehowtogetridoftheomnipresent tachyon. In the process, the constructed theory had spacetime supersymmetry. • Scherk and Schwarz, and independently Yoneya, proposed that closed string theory, ′ always having a massless spin-2 particle, naturally describes gravity and that the scale α should be identified with the Planck scale. Moreover, the theory can be defined in four dimensions using the Kaluza–Klein idea, namely considering the extra dimensions to be compact and small. However, thenewbigimpetusforstringtheorycamein1984. Afterageneralanalysisof gauge and gravitational anomalies 2, it was realized that anomaly-free theories in higher dimensions are very restricted. Green and Schwarz showed in 3 that open superstrings in 10dimensions areanomaly-free ifthe gaugegroupisO(32). E ×E wasalso anomaly-free 8 8 but could not appear in open string theory. In 4 it was shown that another string exists in ten dimensions, a hybrid of the superstring and the bosonic string, which can realize the E ×E or O(32) gauge symmetry. 8 8 Since the early eighties, the field ofstring theory hasbeen continuously developing and we will see the main points in the rest of these lectures. The reader is encouraged to look at a more detailed discussion in 5–8. Onemaywonderwhatmakesstringtheorysospecial. Oneofitskeyingredientsisthat itprovidesafinitetheoryofquantumgravity,atleastinperturbationtheory. Toappreciate the difficulties with the quantization of Einstein gravity, we will look at a single-graviton exchange between two particles (Fig. 1a). We will set h =c = 1. Then the amplitude is 2 2 proportionaltoE /M , whereE is the energy ofthe process andM isthe Planck Planck Planck 19 2 mass, M ∼ 10 GeV. It is related to the Newton constant G ∼M . Thus, we Planck N Planck see that the gravitational interaction is irrelevant in the IR (E M ) but strongly Planck relevant in the UV. In particular it implies that the two-graviton exchange diagram (Fig. 1b) is proportional to Z 4 Λ 1 Λ 3 dE E ∼ , (2.7) 4 4 M 0 M Planck Planck whichisstronglyUV-divergent. InfactitisknownthatEinsteingravitycoupledtomatter is non-renormalizable in perturbation theory. Supersymmetry makes the UV divergence softer but the non-renormalizability persists. There are two ways out of this: • There is a non-trivial UV fixed-point that governs the UV behavior of quantum gravity. To date, nobody has managed to make sense out of this possibility. 8a) b) Figure 1: Gravitational interaction between two particles via graviton exchange. • There is new physics at E ∼ M and Einstein gravity is the IR limit of a more Planck general theory, valid at and beyond the Planck scale. You could consider the analogous situationwith theFermitheoryofweakinteractions. There, onehadanon-renormalizable current–current interaction with similar problems, but today we know that this is the IR ± 0 limit of the standard weak interaction mediated by the W and Z gauge bosons. So far, there is no consistent field theory that can make sense at energies beyondM and Planck containsgravity. Stringsprovide precisely atheorythatinduces new physics atthePlanck scale due to the infinite tower of string excitations with masses of the order of the Planck mass and carefully tuned interactions that become soft at short distance. Moreover string theory seems to have all the right properties for Grand Unification, since it produces and unifies with gravity not only gauge couplings but also Yukawa cou- plings. The shortcomings, to date, of string theory as an ideal unifying theory are its numerous different vacua, the fact that there are three string theories in 10 dimensions that look different (type-I, type II and heterotic), and most importantly supersymmetry breaking. Therehasbeensomeprogressrecentlyinthesedirections: thereisgoodevidence 2 that these different-looking string theories might be non-perturbatively equivalent . 3 Classical string theory As in field theory there are two approaches to discuss classical and quantum string theory. One is the first quantized approach, which discusses the dynamics of a single string. The dynamical variables are the spacetime coordinates of the string. This is an approach that is forced to be on-shell. The other is the second-quantized or field theory approach. Here the dynamical variables are functionals of the string coordinates, or string fields, and we can have an off-shell formulation. Unfortunately, although there is an elegant formulation 2 You will find a pedagogical review of these developments at the end of these lecture notes as well as in 9. 9of open string field theory, the closed string field theory approaches are complicated and difficult to use. Moreover the open theory is not complete since we know it also requires thepresence ofclosed strings. Inthese lectures we will followthe first-quantized approach, although the reader is invited to study the rather elegant formulation of open string field theory 11. 3.1 The point particle Before discussing strings, it is useful to look first at the relativistic point particle. We will use the first-quantized path integral language. Point particles classically follow an extremal path when traveling from one point in spacetime to another. The natural action is proportional to the length of the world-line between some initial and final points: Z Z q s τ f 1 μ ν S =m ds =m dτ −η x˙ x˙ , (3.1.1) μν s τ i 0 μ where η = diag(−1,+1,+1,+1). The momentum conjugate to x (τ) is μν δL mx˙ μ √ p =− = , (3.1.2) μ μ 2 δx˙ −x˙ μ and the Lagrange equations coming from varying the action (3.1.1) with respect toX (τ) read mx˙ μ √ ∂ = 0. (3.1.3) τ 2 −x˙ Equation (3.1.2) gives the following mass-shell constraint : 2 2 p +m =0. (3.1.4) The canonical Hamiltonian is given by ∂L μ H = x˙ −L. (3.1.5) can μ ∂x˙ Inserting(3.1.2)into(3.1.5)wecanseethatH vanishesidentically. Thus, theconstraint can (3.1.4) completely governs the dynamics of the system. We can add it to the Hamiltonian using a Lagrange multiplier. The system will then be described by N 2 2 H = (p +m ), (3.1.6) 2m from which it follows that μ N Nx˙ μ μ μ √ x˙ =x ,H= p = , (3.1.7) 2 m −x˙ or 2 2 x˙ =−N , (3.1.8) 10sowearedescribing time-like trajectories. ThechoiceN=1correspondstoachoiceofscale for the parameter τ, the proper time. Thesquarerootin(3.1.1)isanunwantedfeature. Ofcourseforthefreeparticleitisnot a problem, but as we will see later it will be a problem for the string case. Also the action we used above is ill-defined for massless particles. Classically, there exists an alternative action, which does not contain the square root and in addition allows the generalization to the massless case. Consider the following action : Z   1 −2 μ 2 2 S =− dτe(τ) e (τ)(x˙ ) −m . (3.1.9) 2 The auxiliary variablee(τ) can be viewed as an einbein on the world-line. The associated 2 metric would be g =e , and (3.1.9) could be rewritten as ττ Z q 1 ττ 2 S =− dτ detg (g ∂ x·∂ x−m ). (3.1.10) ττ τ τ 2 The action is invariant under reparametrizations of the world-line. An infinitesimal repa- rametrization is given by μ μ μ μ 2 δx (τ) =x (τ +ξ(τ))−x (τ) =ξ(τ)x˙ +O(ξ ). (3.1.11) Varying e in (3.1.9) leads to Z 1 1 μ 2 2 δS = dτ (x˙ ) +m δe(τ). (3.1.12) 2 2 e (τ) Setting δS = 0 gives us the equation of motion fore : √ 1 −2 2 2 2 e x +m =0 → e = −x˙ . (3.1.13) m Varying x gives Z   1 −2 μ μ δS = dτe(τ) e (τ)2x˙ ∂ δx . (3.1.14) τ 2 After partial integration, we find the equation of motion −1 μ ∂ (e x˙ )= 0. (3.1.15) τ Substituting (3.1.13) into (3.1.15), we find the same equations as before (cf. eq. (3.1.3)). If we substitute (3.1.13) directly into the action (3.1.9), we find the previous one, which establishes the classical equivalence of both actions. We will derive the propagator for the point particle. By definition, Z  Z    ′ x(1)=x 1 1 1 ′ μ μ 2 2 hxxi=N DeDx exp (x˙ ) −em dτ , (3.1.16) x(0)=x 2 0 e where we have put τ =0, τ = 1. 0 1 11Under reparametrizations ofthe world-line, the einbein transformsasa vector. Tofirst order, this means δe =∂ (ξe). (3.1.17) τ This is the local reparametrization invariance of the path. Since we are integrating over e, this means that (3.1.16) will give an infinite result. Thus, we need to gauge-fix the re- parametrization invariance (3.1.17). We can gauge-fixe to be constant. However, (3.1.17) now indicates that we cannot fix more. To see what this constant may be, notice that the length of the path of the particle is Z Z q 1 1 L = dτ detg = dτe, (3.1.18) ττ 0 0 so the best we can do is e = L. This is the simplest example of leftover (Teichmu¨ller) parameters after gauge fixing. The e integration contains an integral over the constant mode as well as the rest. The rest is the “gauge volume” and we will throw it away. Also, to make the path integral converge, we rotate to Euclidean timeτ →iτ. Thus, we are left with     Z Z ′ Z ∞ x(1)=x 1 1 1 ′ μ 2 2 hxxi=N dL Dx exp − x˙ +Lm dτ . (3.1.19) 0 x(0)=x 2 0 L Now write μ μ ′μ μ μ x (τ) =x +(x −x )τ +δx (τ), (3.1.20) μ μ where δx (0) = δx (1) = 0. The first two terms in this expansion represent the classical μ path. The measure for the fluctuations δx is Z Z 1 1 2 μ 2 μ 2 kδxk = dτe(δx ) =L dτ(δx ) , (3.1.21) 0 0 so that √ Y μ μ Dx ∼ Ldδx (τ). (3.1.22) τ Then Z Z R ∞ √ ′ 2 Y 1 (x −x) 1 2 μ 2 ˙ ′ μ − −m L/2 − (δx ) 2L 2L 0 hxxi =N dL Ldδx (τ)e e . (3.1.23) 0 τ μ The Gaussian integral involving δx˙ can be evaluated immediately : D Z    R − √ Y 1 2 1 μ 2 1 ˙ μ − (δx ) 2 L 0 Ldδx (τ)e ∼ det − ∂ . (3.1.24) τ L τ 2 We have to compute the determinant of the operator −∂ /L. To do this we will calcu- τ late first its eigenvalues. Then the determinant will be given as the product of all the eigenvalues. To find the eigenvalues we consider the eigenvalue problem 1 2 − ∂ ψ(τ) =λψ(τ) (3.1.25) τ L with the boundary conditions ψ(0) =ψ(1) =0. Note that there is no zero mode problem here because of the boundary conditions. The solution is 2 n ψ (τ) =C sin(nπτ) , λ = , n =1,2,... (3.1.26) n n n L 12and thus   ∞ 2 Y 1 n 2 det − ∂ = . (3.1.27) τ L L n=1 Obviously the determinant is infinite and we have to regularize it. We will use ζ-function 3 regularization in which ∞ ∞ Y Y ′ −1 −ζ(0) 1/2 a −aζ (0) a/2 L =L =L , n =e =(2π) . (3.1.28) n=1 n=1 Adjusting the normalization factor we finally obtain Z ′ 2 ∞ (x −x) 1 D 2 ′ − − −m L/2 2 2L hxxi = dLL e = (3.1.29) D/2 2(2π) 0 (2−D)/2 ′ 1 x−x ′ = K (mx−x). (D−2)/2 D/2 (2π) m ThisisthefreepropagatorofascalarparticleinDdimensions. Toobtainthemorefamiliar expression, we have to pass to momentum space Z D ip·x pi= d xe xi, (3.1.30) Z Z ′ ′ ′ D −ip·x D ′ ip·x ′ hppi = d xe d x e hxxi Z Z ∞ 1 L ′ ′ 2 2 D ′ i(p −p)·x − (p +m ) 2 = d xe dLe (3.1.31) 2 0 1 D ′ = (2π) δ(p−p) , 2 2 p +m just as expected. ′ Here we should make one more comment. The momentum space amplitudehppi can ip·x also be computed directly if we insert in the path integral e for the initial state and ′ −ip·x e for the final state. Thus, amplitudes are given by path-integral averages of the quantum-mechanical wave-functions of free particles. 3.2 Relativistic strings We now use the ideas of the previous section to construct actions for strings. In the case of point particles, the action was proportional to the length of the world-line between some initial point and final point. For strings, it will be related to the surface area of the “world-sheet” swept by the string as it propagates through spacetime. The Nambu-Goto action is defined as Z S =−T dA. (3.2.1) NG 3 You will find more details on this in 13. 13−2 The constant factor T makes the action dimensionless; its dimensions must be length 2 i or mass . Suppose ξ (i =0,1) are coordinates on the world-sheet andG is the metric μν of the spacetime in which the string propagates. Then, G induces a metric on the μν world-sheet : μ ν ∂X ∂X 2 μ ν i j i j ds =G (X)dX dX =G dξ dξ =G dξ dξ , (3.2.2) μν μν ij i j ∂ξ ∂ξ where the induced metric is μ ν G =G ∂X ∂ X . (3.2.3) ij μν i j This metric can be used to calculate the surface area. If the spacetime is flat Minkowski space then G =η and the Nambu-Goto action becomes μν μν Z Z q q 2 2 ′ 2 2 ′2 ˙ ˙ S =−T −detG d ξ =−T (X.X ) −(X )(X )d ξ, (3.2.4) NG ij μ μ μ ∂X ′μ ∂X 0 1 ˙ where X = and X = (τ =ξ , σ =ξ ). The equations of motion are ∂τ ∂σ δL δL ∂ +∂ =0. (3.2.5) τ σ ′μ ˙μ δX δX Depending on the kind of strings, we can impose different boundary conditions. In the case of closed strings, the world-sheet is a tube. If we let σ run from 0 to σ¯ = 2π, the boundary condition is periodicity μ μ X (σ+σ¯) =X (σ). (3.2.6) For open strings, the world-sheet is a strip, and in this case we will putσ¯ =π. Two kinds 4 of boundary conditions are frequently used : • Neumann : δL =0; (3.2.7) ′μ δX σ=0,σ¯ • Dirichlet : δL = 0. (3.2.8) ˙μ δX σ=0,σ¯ As we shall see at the end of this section, Neumann conditions imply that no momentum flows off the ends of the string. The Dirichlet condition implies that the end-points of the string are fixed in spacetime. We will not discuss them further, but they are relevant for describing (extended) solitons in string theory also known as D-branes 10. μ The momentum conjugate to X is ′ ′μ ′ 2 μ ˙ ˙ δL (X·X )X −(X ) X μ Π = =−T . (3.2.9) ˙μ ′ ˙ 2 ˙ 2 ′ 2 1/2 δX (X ·X) −(X) (X ) 4 One could also impose an arbitrarylinear combination of the two boundary conditions. We will come back to the interpretation of such boundary conditions in the last chapter. 142 δ L μ ′μ ˙ The matrix has two zero eigenvalues, with eigenvectors X and X . This signals ˙μ ˙ν δX δX the occurrence of two constraints that follow directly from the definition of the conjugate momenta. They are ′ 2 2 ′2 Π·X = 0 , Π +T X = 0. (3.2.10) The canonical Hamiltonian Z σ¯ ˙ H = dσ(X·Π−L) (3.2.11) 0 vanishes identically, just in the case of the point particle. Again, the dynamics is governed solely by the constraints. ThesquarerootintheNambu-Gotoactionmakesthetreatmentofthequantumtheory quitecomplicated. Again,wecansimplifytheactionbyintroducinganintrinsicfluctuating metric on the world-sheet. In this way, we obtain the Polyakov action for strings moving in flat spacetime 12 Z q T 2 αβ μ ν S =− d ξ −detg g ∂ X ∂ X η . (3.2.12) P α β μν 2 As is well known from field theory, varying the action with respect to the metric yields the stress-tensor : 2 1 δS P 1 γδ √ T ≡− =∂ X·∂ X− g g ∂ X·∂ X. (3.2.13) αβ α β αβ γ δ 2 αβ T −detgδg Setting this variation to zero and solving forg , we obtain, up to a factor, αβ g =∂ X·∂ X. (3.2.14) αβ α β In other words, the world-sheet metric g is classically equal to the induced metric. If αβ we substitute this back into the action, we find the Nambu-Goto action. So both actions are equivalent, at least classically. Whether this is also true quantum-mechanically is not clear in general. However, they can be shown to be equivalent in the critical dimension. From now on we will take the Polyakov approach to the quantization of string theory. μ By varying (3.2.12) with respect to X , we obtain the equations of motion: q 1 αβ μ √ ∂ ( −detgg ∂ X )= 0. (3.2.15) α β −detg Thus,theworld-sheetactioninthePolyakovapproachconsistsofDtwo-dimensionalscalar μ fields X coupled to the dynamical two-dimensional metric and we are thus considering a theory of two-dimensional quantum gravity coupled to matter. One could ask whether there are other terms that can be added to (3.2.12). It turns out that there are only two: the cosmological term Z q λ −detg (3.2.16) 1 and the Gauss-Bonnet term Z q (2) λ −detgR , (3.2.17) 2 15(2) where R is the two-dimensional scalar curvature associated with g . This gives the αβ Euler number of the world-sheet, which is a topological invariant. So this term cannot influence the local dynamics of the string, but it will give factors that weight various topologies differently. It is not difficult to prove that (3.2.16) has to be zero classically. In fact the classical equations of motion for λ 6= 0 imply that g = 0, which gives trivial 1 αβ dynamics. We will not consider it further. For the open string, there are other possible terms, which are defined on the boundary of the world-sheet. We will discuss the symmetries of the Polyakov action: • Poincar´e invariance : μ μ ν μ δX =ω X +α , δg = 0, (3.2.18) αβ ν where ω =−ω ; μν νμ • local two-dimensional reparametrization invariance : γ γ γ δg = ξ ∂ g +∂ ξ g +∂ ξ g =∇ ξ +∇ ξ , αβ γ αβ α βγ β αγ α β β α μ α μ δX = ξ ∂ X , α q q α δ( −detg) = ∂ (ξ −detg); (3.2.19) α • conformal (or Weyl) invariance : μ δX =0 , δg =2Λg . (3.2.20) αβ αβ Due to the conformal invariance, the stress-tensor will be traceless. This is in fact true i in general. Consider an action S(g ,φ) in arbitrary spacetime dimensions. We assume αβ that it is invariant under conformal transformations i i δg = 2Λ(x)g , δφ =d Λ(x)φ . (3.2.21) αβ αβ i The variation of the action under infinitesimal conformal transformations is " Z X δS δS 2 αβ 0 =δS = d ξ 2 g + d φ Λ. (3.2.22) i i αβ δg δφ i i δS Using the equations of motion for the fields φ, i.e. =0, we find i δφ i δS α αβ T ∼ g = 0, (3.2.23) α αβ δg which follows without the use of the equations of motion, if and only if d = 0. This is i the case for the bosonic string, described by the Polyakov action, but not for fermionic extensions. 16Just as we could fix e(τ) for the point particle using reparametrization invariance, we can reduce g to η = diag(−1,+1). This is called conformal gauge. First, we choose a αβ αβ parametrization that makes the metric conformally flat, i.e. 2Λ(ξ) g =e η . (3.2.24) αβ αβ Itcanbeproventhatintwodimensions, thisisalwayspossibleforworld-sheetswithtrivial topology. We will discuss the subtle issues that appear for non-trivial topologies later on. Using the conformal symmetry, we can further reduce the metric toη . We also work αβ with “light-cone coordinates” ξ =τ +σ , ξ =τ−σ. (3.2.25) + − The metric becomes 2 ds =−dξ dξ . (3.2.26) + − The components of the metric are 1 g =g = 0 , g =g =− (3.2.27) ++ −− +− −+ 2 and 1 ∂ = (∂ ±∂ ). (3.2.28) τ σ ± 2 The Polyakov action in conformal gauge is Z 2 μ ν S ∼T d ξ ∂ X ∂ X η . (3.2.29) P + − μν By going to conformal gauge, we have not completely fixed all reparametrizations. In particular, the reparametrizations ξ −→f(ξ ) , ξ −→g(ξ ) (3.2.30) + + − − only put a factor ∂ f∂ g in front of the metric, so they can be compensated by the + − 2 transformation of d ξ. Notice that here we have exactly enough symmetry to completely fix the metric. A metric on a d-dimensional world-sheet has d(d+1)/2 independent components. Using reparametrizations, d of them can be fixed. Conformal invariance fixes one more compo- nent. The number of remaining components is d(d+1) −d−1. (3.2.31) 2 This is zero in the case d = 2 (strings), but not for d 2 (membranes). This makes an analogous treatment of higher-dimensional extended objects problematic. We will derive the equations of motion from the Polyakov action in conformal gauge μ (eq. (3.2.29)). By varying X , we get (after partial integration): Z Z τ 1 2 μ ′ μ δS =T d ξ(δX ∂ ∂ X )−T dτX δX . (3.2.32) + − μ μ τ 0 17Using periodic boundary conditions for the closed string and ′μ X =0 (3.2.33) σ=0,σ¯ for the open string, we find the equations of motion μ ∂ ∂ X = 0. (3.2.34) + − Even after gauge fixing, the equations of motion for the metric have to be imposed. They are T = 0, (3.2.35) αβ or 1 ′ 2 ′2 1 ˙ ˙ T =T = X·X =0 , T =T = (X +X ) =0, (3.2.36) 10 01 00 11 2 4 which can also be written as ′ 2 ˙ (X±X ) = 0. (3.2.37) These are known as the Virasoro constraints. They are the analog of the Gauss law in the string case. In light-cone coordinates, the components of the stress-tensor are 1 1 T = ∂ X·∂ X , T = ∂ X·∂ X , T =T = 0. (3.2.38) ++ + + −− − − +− −+ 2 2 α This last expression is equivalent to T = 0; it is trivially satisfied. Energy-momentum α α conservation,∇ T = 0, becomes αβ ∂ T +∂ T =∂ T +∂ T = 0. (3.2.39) − ++ + −+ + −− − +− Using (3.2.38), this states ∂ T =∂ T = 0 (3.2.40) − ++ + −− which leads to conserved charges Z σ¯ + + Q = f(ξ )T (ξ ), (3.2.41) f ++ 0 and likewise for T . To convince ourselves that Q is indeed conserved, we need to −− f calculate Z σ¯ + + 0= dσ∂ (f(ξ )T )=∂ Q + f(ξ )T . (3.2.42) − ++ τ f ++ 0 For closed strings, the boundary term vanishes automatically; for open strings, we need to use the constraints. Of course, there are other conserved charges in the theory, namely those associated with Poincar´e invariance : q αβ α P =−T detgg ∂ X , (3.2.43) β μ μ q α αβ J =−T detgg (X ∂ X −X ∂ X ). (3.2.44) μ β ν ν β μ μν 18α α We have ∂ P = 0 = ∂ J because of the equation of motion for X. The associated α α μ μν charges are Z Z σ¯ σ¯ τ τ P = dσP , J = dσJ . (3.2.45) μ μν μ μν 0 0 These are conserved, e.g. Z Z σ¯ σ¯ ∂P μ 2 2 = T dσ∂ X =T dσ∂ X μ μ τ σ ∂τ 0 0 = T(∂ X (σ =σ¯)−∂ X (σ = 0)). (3.2.46) σ μ σ μ (In the second line we used the equation of motion forX.) This expression automatically vanishes for the closed string. For open strings, we need Neumann boundary conditions. Here we see that these conditions imply that there is no momentum flow off the ends of the string. The same applies to angular momentum. 3.3 Oscillator expansions We will now solve the equations of motion for the bosonic string, μ ∂ ∂ X = 0, (3.3.1) + − taking into account the proper boundary conditions. To do this we have to treat the open and closed string cases separately. We will first consider the case of the closed string. • Closed Strings The most general solution to equation (3.3.1) that also satisfies the periodicity con- dition μ μ X (τ,σ+2π)=X (τ,σ) can be separated in a left- and a right-moving part: μ μ μ X (τ,σ)=X (τ +σ)+X (τ−σ), (3.3.2) L R where μ μ μ X x p i α¯ μ k −ik(τ+σ) √ X (τ +σ)= + (τ +σ)+ e , L 2 4πT k 4πT k6=0 (3.3.3) μ μ μ X x p i α μ k −ik(τ−σ) X (τ−σ)= + (τ−σ)+√ e . R 2 4πT k 4πT k6=0 μ μ The α and α¯ are arbitrary Fourier modes, and k runs over the integers. The k k μ μ μ functionX (τ,σ) must be real, so we know thatx andp must also be real and we can derive the following reality condition for the α’s: μ ∗ μ μ ∗ μ (α ) =α and (α¯ ) =α¯ (3.3.4) k −k k −k 19

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